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Theorem elssetkg 4269
 Description: Membership via the Kuratowski subset relationship. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
elssetkg ((A V B W) → (⟪{A}, B SkA B))

Proof of Theorem elssetkg
StepHypRef Expression
1 snex 4111 . . 3 {A} V
2 opkelssetkg 4268 . . 3 (({A} V B W) → (⟪{A}, B Sk ↔ {A} B))
31, 2mpan 651 . 2 (B W → (⟪{A}, B Sk ↔ {A} B))
4 snssg 3844 . . 3 (A V → (A B ↔ {A} B))
54bicomd 192 . 2 (A V → ({A} BA B))
63, 5sylan9bbr 681 1 ((A V B W) → (⟪{A}, B SkA B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  {csn 3737  ⟪copk 4057   Sk cssetk 4183 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058  df-ssetk 4193 This theorem is referenced by:  elssetk  4270  opkelimagekg  4271  setswith  4321
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